Compact metrizable groups are isometry groups of compact metric spaces

This note is devoted to proving the following result: given a compact metrizable group , there is a compact metric space such that is isomorphic (as a topological group) to the isometry group of .

     

Properly embedded least area planes in Gromov hyperbolic -spaces

Let be a Gromov hyperbolic -space with cocompact metric, and the sphere at infinity of . We show that for any simple closed curve in , there exists a properly embedded least area plane in spanning . This gives a positive answer to Gabai's conjecture from 1997. Soma has already proven this conjecture in 2004. Our technique here is simpler and more general, and it can be applied to many similar settings.

     

Algebraic characterizations of measure algebras

We present necessary and sufficient conditions for the existence of a countably additive measure on a Boolean -algebra. For instance, a Boolean -algebra is a measure algebra if and only if is the union of a chain of sets such that for every ,

(i)

every antichain in has at most elements (for some integer ),

(ii)

if is a sequence with for each , then , and

(iii)

for every , if is a sequence with , then for eventually all , .

The chain is essentially unique.

     

Subgroups generated by small classes in finite groups

Let be the subgroup of generated by all elements that lie in conjugacy classes of the two smallest sizes. Avinoam Mann showed that if is nilpotent, then has nilpotence class at most . Using a slight variation on Mann's methods, we obtain results that do not require us to assume that is nilpotent. We show that if is supersolvable, then is nilpotent with class at most , and in general, the Fitting subgroup of has class at most .

     

A group structure on squares

We show that there is an abelian group structure on the orbit set of ``squares' of unimodular rows of length over a commutative ring of stable dimension when , odd and also an abelian group structure on the orbit set of ``fourth powers' of unimodular rows of length over a commutative ring of stable dimension when , even.

     

A remark on irregularity of the -Neumann problem on non-smooth domains

It is an observation due to J. J. Kohn that for a smooth bounded pseudoconvex domain in there exists such that the -Neumann operator on maps (the space of -forms with coefficient functions in -Sobolev space of order ) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain in , smooth except at one point, whose -Neumann operator is not bounded on for any .

     

On Faltings' annihilator theorem

In the present article, the author shows that Faltings' annihilator theorem holds for any Noetherian ring if is universally catenary; all the formal fibers of all the localizations of  are Cohen-Macaulay; and the Cohen-Macaulay locus of each finitely generated -algebra is open.

     

Julia sets converging to the unit disk

We consider the family of rational maps , where and is small. If is equal to 0, the limiting map is and the Julia set is the unit circle. We investigate the behavior of the Julia sets of when tends to 0, obtaining two very different cases depending on and . The first case occurs when ; here the Julia sets of converge as sets to the closed unit disk. In the second case, when one of or is larger than , there is always an annulus of some fixed size in the complement of the Julia set, no matter how small is.

     

Rademacher bounded families of operators on

We give a characterization of R-bounded families of operators on We then use this result to study sectorial operators on . We show that if is an R-sectorial operator on , then, for any there is an invertible operator with such that for some strictly positive Borel function , contains the weighted -space

     

The -primary classifying space of the fiber of the double suspension

Gray showed that the homotopy fiber of the double suspension has an integral classifying space , which fits in a homotopy fibration . In addition, after localizing at an odd prime , is an -space and if , then is homotopy associative and homotopy commutative, and is an -map. We positively resolve a conjecture of Gray's that the same multiplicative properties hold for as well. We go on to give some exponent consequences.

     

-actions with -dimensional fixed point set

We describe the equivariant cobordism classification of smooth actions of the group , considered as the group generated by two commuting involutions, on closed smooth -dimensional manifolds , for which the fixed point set of the action is a connected manifold of dimension and or . For , the classification is known.

     

On Generic differential -extensions

Let be an algebraically closed field with trivial derivation and let denote the differential rational field , with , , , , differentially independent indeterminates over . We show that there is a Picard-Vessiot extension for a matrix equation , with differential Galois group , with the property that if is any differential field with field of constants , then there is a Picard-Vessiot extension with differential Galois group if and only if there are with well defined and the equation giving rise to the extension .

     

Groups which do not admit ghosts

A ghost in the stable module category of a group is a map between representations of that is invisible to Tate cohomology. We show that the only non-trivial finite -groups whose stable module categories have no non-trivial ghosts are the cyclic groups and . We compare this to the situation in the derived category of a commutative ring. We also determine for which groups the second power of the Jacobson radical of is stably isomorphic to a suspension of .

     

A Banach-Stone theorem for Riesz isomorphisms of Banach lattices

Let and be compact Hausdorff spaces, and , be Banach lattices. Let denote the Banach lattice of all continuous -valued functions on equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism such that is non-vanishing on if and only if is non-vanishing on , then is homeomorphic to , and is Riesz isomorphic to . In this case, can be written as a weighted composition operator: , where is a homeomorphism from onto , and is a Riesz isomorphism from onto for every in . This generalizes some known results obtained recently.

     

A lower bound for the equilateral number of normed spaces

We show that if the Banach-Mazur distance between an -dimensional normed space and is at most , then there exist equidistant points in . By a well-known result of Alon and Milman, this implies that an arbitrary -dimensional normed space admits at least equidistant points, where is an absolute constant. We also show that there exist equidistant points in spaces sufficiently close to , .

     

Detecting completeness from Ext-vanishing

Motivated by work of C. U. Jensen, R.-O. Buchweitz, and H. Flenner, we prove the following result. Let be a commutative noetherian ring and an ideal in the Jacobson radical of . Let be the -adic completion of . If is a finitely generated -module such that for all , then is -adically complete.

     

Extreme points, exposed points, differentiability points in CL-spaces

This paper presents a property of geometric and topological nature of Gateaux differentiability points and Fréchet differentiability points of almost CL-spaces. More precisely, if we denote by a maximal convex set of the unit sphere of a CL-space , and by the cone generated by , then all Gateaux differentiability points of are just n-s, and all Fréchet differentiability points of are (where n-s denotes the non-support points set of ).

     

On the norm of an idempotent Schur multiplier on the Schatten class

We show that if the norm of an idempotent Schur multiplier on the Schatten class lies sufficiently close to , then it is necessarily equal to . We also give a simple characterization of those idempotent Schur multipliers on whose norm is .

     

Uniqueness of structures for connective covers

We refine our earlier work on the existence and uniqueness of structures on -theoretic spectra to show that the connective versions of real and complex -theory as well as the connective Adams summand at each prime have unique structures as commutative -algebras. For the -completion we show that the McClure-Staffeldt model for is equivalent as an ring spectrum to the connective cover of the periodic Adams summand . We establish a Bousfield equivalence between the connective cover of the Lubin-Tate spectrum and .